Theorem negm 8550

Description: The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)

Assertion

Ref	Expression
negm	|- ( ( A C_ RR /\ E. x x e. A ) -> E. y y e. { z e. RR | -u z e. A } )

Distinct variable group:   x, A, y, z

Proof of Theorem negm

Step	Hyp	Ref	Expression
1		ssel 2939	. . . . 5 |- ( A C_ RR -> ( x e. A -> x e. RR ) )
2		renegcl 7272	. . . . . . . 8 |- ( x e. RR -> -u x e. RR )
3		negeq 7204	. . . . . . . . . 10 |- ( z = -u x -> -u z = -u -u x )
4	3	eleq1d 2106	. . . . . . . . 9 |- ( z = -u x -> ( -u z e. A <-> -u -u x e. A ) )
5	4	elrab3 2699	. . . . . . . 8 |- ( -u x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> -u -u x e. A ) )
6	2, 5	syl 14	. . . . . . 7 |- ( x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> -u -u x e. A ) )
7		recn 7014	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
8	7	negnegd 7313	. . . . . . . 8 |- ( x e. RR -> -u -u x = x )
9	8	eleq1d 2106	. . . . . . 7 |- ( x e. RR -> ( -u -u x e. A <-> x e. A ) )
10	6, 9	bitrd 177	. . . . . 6 |- ( x e. RR -> ( -u x e. { z e. RR | -u z e. A } <-> x e. A ) )
11	10	biimprd 147	. . . . 5 |- ( x e. RR -> ( x e. A -> -u x e. { z e. RR | -u z e. A } ) )
12	1, 11	syli 33	. . . 4 |- ( A C_ RR -> ( x e. A -> -u x e. { z e. RR | -u z e. A } ) )
13		elex2 2570	. . . 4 |- ( -u x e. { z e. RR | -u z e. A } -> E. y y e. { z e. RR | -u z e. A } )
14	12, 13	syl6 29	. . 3 |- ( A C_ RR -> ( x e. A -> E. y y e. { z e. RR | -u z e. A } ) )
15	14	exlimdv 1700	. 2 |- ( A C_ RR -> ( E. x x e. A -> E. y y e. { z e. RR | -u z e. A } ) )
16	15	imp 115	1 |- ( ( A C_ RR /\ E. x x e. A ) -> E. y y e. { z e. RR | -u z e. A } )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   {crab 2310   C_ wss 2917   RRcr 6888   -ucneg 7183

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-neg 7185

This theorem is referenced by: (None)